A double category has an equivalent description as a double partial monoid, that is, a set equipped with two partial monoid structures such that all structure maps are partial monoid homomorphisms. Double categories can have an interesting and natural involutive structure, which we characterise in terms of ``internal adjointability'' functors (or partial monoid homomorphisms). There is a well-known result by Brown, Mosa and Spencer, demonstrating the equivalence of categories between the category of edge symmetric double categories with connection and the category of 2-categories. (A 2-category is equivalent to a double partial monoid with an extra condition.) With the intrinsic opposite categorical structures taken into account, the situation is more complex but more symmetrical, and we arrive at a new result involving an isomorphism of categories, (a stronger equivalence). (For people familiar with connections and/or conjoints, conjoint data is equivalent to the connection data together with that of horizontal and vertical 2-arrow involutions, for instance: $\Gamma^{*^2_v}$ and $\Gamma^{*^2_h}$.) Involutive double categories are good generalisations of symmetric monoidal categories with duals. Co-authors: Paolo Bertozzini and Roberto Conti (with helpful input also from Pedro Resende).